J . Am. Chem. SOC.1994,116, 9644-9651
9644
Trajectory Studies of s N 2 Nucleophilic Substitution. 4. Intramolecular and Unimolecular Dynamics of the Cl----CH3Br and ClCH3---Br- Complexes Haobin Wang, Cilles H. Pesherbe, and William L. Hase’ Contribution from the Department of Chemistry, Wayne State University, Detroit, Michigan 48202 Received May 12, 1994”
Abstract: Classical trajectory calculations, performed on an analytic potential energy function derived from ab initio calculations, are used to study the intramolecular and unimolecular dynamics of the Cl----CH3Br complex with initial mode specific excitation. Two distinct patterns are observed in the dynamics of this complex. When the low-frequency modes are excited, the complex preferentially dissociates to C1- CH3Br. However, when the high-frequency CH3Br ClCH3---Br intramolecular modes are excited, the above is a negligible reaction path and, instead, Cl----CH3Br becomes important. Contrary to R R K M theory, the ClCH3---Br complexes formed by this isomerization do not immediately dissociate to ClCH3 B r but remain trapped in the central barrier region of the potential energy surface, with extensive barrier recrossings. The intramolecular dynamics of C1----CH3Br and ClCH3---Br are interpreted in terms of intermolecular and intramolecular complexes, with the former accessing the dissociation products and the latter the central barrier region. There is a dynamical bottleneck for transitions between these two complexes. The ClCH3 Br product energies, for ClCH3---Br complexes which do dissociate, are in agreement with the previous experimental study of Graul and Bowers [ J . Am. Chem. SOC.1991, 113, 96961.
-
+
+
+
I. Introduction Recently, there has been considerable interest in the dynamics and kinetics of gas-phase s N 2 nucleophilic substitution reaction^.*-*^ These reactions are characterized by a potential energy surface with a central barrier which separates pre- and Abstract published in Advance ACS Abstracts, September 15, 1994. (1) Barlow, S. E.: Van Doren. J. M.: Bierbaum. V. M. J . Am. Chem. SOC. 1988.110.7240. (2) Van Doren, J. M.; Depuy, C. H.; Bierbaum, V. M. J . Phys. Chem. 1989,93,1130. (3) DePuy, C. H.; Gronert, S.; Mullin, A.; Bierbaum, V. M. J. Am. Chem. SOC. 1990,i12,8650. (4) Gronert.. S.:. DeDuv. . . C. H.: Bierbaum. V. M. J . Am. Chem. SOC.1991, 11 3.4009. ( 5 ) Viggiano, A. A.; Morris, R. A.; Dale, F.; Pauson, J. F.; Giles, K.; Smith, D.; Su, T. J . Chem. Phys. 1990,93,1149. (6) Viggiano, A. A.; Paschkewitz, J. S.; Morris, R. A.; Paulson, J. F.; Gonzalez-Lafont, A.; Truhlar, D. G. J . Am. Chem. SOC.1991, 113,9404. (7) Viggiano, A. A,; Morris, R. A.; Paschkewitz, J. S.; Paulson, J. F. J . Am. Chem. SOC.1992,114,10477. (8) (a) Graul, S. T.; Bowers, M. T. J . Am. Chem. SOC.1991,113,9696. (b) Graul, S.T.; Bowers, M. T. J . Am. Chem. SOC.1994,116,3875. (9) Cyr, D. M.; Posey, L. A,; Bishea, G. A.; Han, C.-C.; Johnson, M. A. J. Am. Chem. SOC.1991,113,9697. (10) Cyr, D. M.; Bishea, G. A.; Scarton, M. G.; Johnson, M. A. J . Chem. Phys. 1992,97,5911. (1 1) Wladkowski, B. D.; Lim, K. F.; Allen, W. D.; Brauman, J. I. J. Am. Chem. SOC.1992,114,9136. (12) Giles, K.; Grimsrud, E. P. J . Phys. Chem. 1992,96,6680. (13) Strode, K. S.;Grimsrud, E. P. Submitted for publication. (14) Knighton, W. B.;Bognar, J. A.; OConnor, P. M.; Grimsrud, E. P. J . Am. Chem. SOC.1993,115, 12079. (15) Tucker, S. C.; Truhlar, D. G. J . Phys. Chem. 1989,93,8138. (16) Tucker, S. C.; Truhlar, D. G. J. Am. Chem. SOC.1990,112,3338. (17) Tucker, S. C.; Truhlar, D. G. J . Am. Chem. SOC.1990,112,3347. (18) Vande Linde, S.R.; Hase, W. L. J. Am. Chem. SOC.1989,llI , 2349. (19) Vande Linde, S. R.; Hase, W. L. J . Phys. Chem. 1990,94,2778. (20) Vande Linde, S. R.; Hase, W. L. J . Phys. Chem. 1990,94,6148. (21) Vande Linde, S. R.; Hase, W. L. J. Chem. Phys. 1990,93,7962. (22) Cho, Y. J.; Vande Linde, S. R.; Zhu, L.; Hase, W. L. J. Chem. Phys. @
1992. - - - -, -96. -, 8275. -- . - .
(23) Hase, W. L.; Cho, Y. J. J. Chem. Phys. 1993,98,8626. (24) Billing, G. D. Chem. Phys. 1992,159, 109. (25) Basilevsky, M. V.; Ryaboy, V. M. Chem. Phys. Lett. 1986,129,71. (26) Ryaboy, V. M. Chem. Phys. Lett. 1989,159,371. (27) Ryaboy, V. M. In Dynamics of Ion-Molecule Complexes; Hase, W. L., Ed.; Advances in Classical Trajectory Methods, Vol. 2.; JAI Press: Greenwich, CT, 1993.
postreaction ion-dipole complexes. The model proposed by Brauman and c o - ~ o r k e r s ? ” ~in~which an ion-molecule capture theory33 is used to calculate the association rate constant for forming the prereaction complex and Rice-Ramsperger-KasselMarcus (RRKM) theory”38 is used to calculate lifetimes for the complex, has been very successful in interpreting the kinetics of many gas-phase s N 2 reactions.’ However, recent experimenthat this statistical model t a l s ~and ~ . ~theoreti~a1~”~~workindicates may be incomplete and dynamics may have to be included for some s N 2 reactions. One S Nreaction, ~ for which experimental results appear to be inconsistent with the above statistical model, is C1-
+ CH3Br
-
ClCH,
+ Br-
(1)
Graul and Bowers8 found relative translational energies between the reaction products ClCH3 + B r to be much smaller than the prediction of orbiting transition state/phase space theory (OTS/ PST),39 which assumes statistical energy partitioning a t the orbiting transition statem for dissociation of the postreaction complex ClCH3---Bra Viggiano et al.7 measured the rate of (28) Pellerite, M.J.; Brauman, J. I. J. Am. Chem. SOC.1980,102,5993. (29) Pellerite, M. J.; Brauman, J. I. J. Am. Chem. SOC.1983,105,2672. (30)Dodd, J. A.; Brauman, J. I. J. Am. Chem. SOC.1984,106,5356. (31) Dodd, J. A.; Brauman, J. I. J. Phys. Chem. 1986,90,3559. (32) Moylan, C. R.; Brauman, J. I. In Dynamics of Ion-Molecule Complexes; Hase, W. L., Ed.; Advances in Classical Trajectory Methods, Vol. 2.; JAI Press: Greenwich, CT, 1993. (33) Hase, W.L.; Wardlaw, D. M. In Biomolecular Collisions;Ashfold, M. N. R., Baggot, J. E., Ed.; Royal Society of Chemistry: London, 1989. (34) Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973. (35) Robinson, P. J.; Holbrook, K. A. UnimolecularReactions; John Wiley & Sons: New York, 1972; p 128. (36) Steinfeld, J. I.; Francisco, J. S.; Hase, W. L. Chemicul Kinetics and Dynamics; Prentice Hall: Englewood Cliffs, NJ, 1989. (37) Gilbert, R. G.; Smith, S. C. Theory of Unimolecular and Recombination Reactions; Blackwell: Boston, 1990. (38) Hase, W. L. In Dynamics of Molecular Collisions;Miller, W. H., Ed.; Plenum: New York, 1976; Part B, p 121. (39) Light, J. C. Discuss. Faraday SOC.1967,44,14.Klots, C. E. J . Phys. Chem. 1971,75,1526. Klots, C. E. Z . Naturforsch., Teil A 1972,27,553. (40)Chesnavich,W. J.; Bowers, M. T. I? Gas Phase Ion Chemistry; Bowers, M. T., Ed.; Academic Press: New York, 1979.
0002-786319411516-9644$04.50/0 0 1994 American Chemical Society
J. Am. Chem. Soc., Vol. 116, No. 21, 1994 9645
Trajectory Studies of S,2 Nucleophilic Substitution Table 1. Geometries and Energies for Stationary Points on PESl(BrP
CI- + CH3Br reactants
internal coordinates rc-a rc-Br
Cl----CHnBr complex 3.221 1.991 1.07 1 72.9 107.1 111.5 -10.731
m
1.944 1.077 72.4 107.6 111.3
rC-H
bHX-Cl bH-C-Br hGn potential energy
[Cl---CH3---Br]saddlepoint
0
ClCHs + B r products
ClCHs---Br complex
2.470 2.462 1.062 87.4 92.6 119.8 -2.785
1.819 3.527 1.074 108.1 71.9 11 1.8 -2 1.209
1.789 m
1.077 108.1 71.9 110.8 -12.647
Distances are in A, angles are in deg, and energies are in kcal/mol. Table 2. Harmonic Vibrational Frequencies for Stationary Points on PESl(Br)'
vibrational mode AI, C-H str AI, CH3 def AI, C-Br strb AI, C-CI strb E, C-H str E, CH3 def E, CHI rock E, CI- (or B r ) bendsc
CI- + CH3Br reactants
CI----CH3Br complex
[Cl---CHp---Br]saddlepoint
CICH3---Br complex
3048 1497 620
3147d 1467
500
3215 1185 161
91 3293 1452 1089 72
3415 1340 1155 169
3101 1548 68 646 3238 1462 1123 64
3183 1457 1065
400i
CICH3 + B r Droducts 3050 1550 739 3181 1460 1108
a The vibrational frequencies are in units of cm-l. At the saddlepoint the C-Br and C-CI stretches become a symmetric C1-C-Br stretch and an asymmetric C1-C-Br stretch with an imaginary frequency. This mode is a CI-C-Br bend at the saddlepoint. d This frequency is given incorrectly as 3047 cm-I in ref 41.
reaction 1 versus reagent relative translational energy and CH3Br temperature. For a fixed relative translational energy, they found the rate constant does not depend on the temperature of CH3Br, a result inconsistent with R R K M theory. This is because R R K M theory predicts that the ratio of the rate constants for Cl----CH3Br decomposition to C1- CH3Br and ClCH3 B r varies with CH3Br temperature (i.e., vibrational/rotational energy). The rate of reaction 1 has been measured versus pressure,14 which provides insight into the lifetime of the Cl----CH3Br complex. In the work reported here, a multidimensional analytic potential energy function, recently developed for reaction 1,4l is used in a classical trajectory study of the intramolecular and unimolecular dynamics associated with the Cl----CH3Br and ClCH3---Br complexes. Trajectories are initialized in the Cl----CH3Br complex region of the potential energy surface with a total energy approximately characteristic of complexes formed by C1- CH3Br collisions a t 300 K. Different internal energy distributions are considered for the Cl----CHSBr complex to determine whether the nature of the modes initially excited affects the chemical dynamics. Determined from the trajectories are the lifetimes of the Cl----CH3Br and ClCHs---Br complexes, the importance of central barrier recrossing, and the ClCH3 B r product energies.
+
+
0.0
-5.0
-10.0
-15.0
-20.0
-25.0
-10.0 -8.0 -6.0
+
+
-
An analytic potential energy function developed previously for the C1- CH3Cl ClCH3 C1- reactionI9 was modified to derive a multidimensional analytic function for reaction 1.41 Every atom and degree of freedom is treated explicitly by this latter potential function, which is written as Vtotal
=
VCl(ra*ga)
V@(ra*ga)
+
[ - sLR(ga)l - sLR(ga)l
+ VBr(rb,gb)
[ - sLR(gb)l
+ V&,(rb,gb) [ - sLR(&)l
+
+
where ra = rczI, r b = rC-Br, g, = ra - rb, and w = r b - ra. The , and Vc1er terms for shortpotential consists of the Vel, V B ~V,, range interactions, the V, and VHC terms for the CH3 moiety, (41) Wang, H.; Zhu,
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
gb (A) Figure 1. Reactionpathpotentialversusgb = rc-Br-rwlforthepotential energy surface used in this work. Zero-point energy is not included in the potential energy curve. For stationary points on the potential energy surface the harmonic zero-point energy is as follows: C1- + CH3Br reactants, 23.70 kcal/mol; Cl----CHoBr complex, 24.33 kcal/mol; [CI---CH3---Br]-saddle point, 23.90 kcal/mol; ClCH3---Br complex, 24.50 kcal/mol; ClCH3 B r products, 24.07 kcal/mol.
+
11. Potential Energy Function
+
-4.0
L.;Hase, W. L. J . Phys. Chem. 1994, 98, 1608.
cR
eR
for the C1- + CH3Br long-range interactions, and for theClCH3 B r long-range interactions. Twosets of parameters were chosen for this analytic potential?' by fitting ab initio calculations and experimental vibrational frequencies, structures, and energies to give two potential energy surfaces identified as PESl(Br) and PES2(Br). The two surfaces only differ in their values for the H-C-Cl and H-C-Br bending force constants.41 PES 1(Br) is used for the trajectory calculations reported here. PES 1(Br) has been described in considerable detail previous191 and only several essential features are reviewed here. Geometries and energies for stationary points on the potential energy function are listed in Table 1. Harmonic vibrational frequencies for these stationary points are listed in Table 2. The reaction path potential for PESl(Br) was determined by following the paths of steepest descent in mass-weighted Cartesian coordinates.41 In Figure 1 the reaction path potential is plotted versus g b = f G B r - rc-cl.
+
Wang et al.
9646 J . Am. Chem. SOC.,Vol. 116, No. 21, 1994 111. Classical Trajectory Calculations
ThePESl(Br) potential energy functiondescribedaboveandderivatives of this potential with respect to Cartesian coordinates were incorporated into the general chemical dynamics computer program VENUS,'2 which was used for the classical trajectory calculations reported here. The trajectories were started from the prereaction complex Cl----CHpBr to model experiments by Graul and Bowers8 and Cyr et aL9 The algorithms used to select initial conditions for the complex are standard options in VENUS and have been described in considerable detail el~ewhere.4~ In the trajectory calculations different initial non-random energy distributions were investigated for the CI----CHpBr complex at a total energy approximately 2.7 kcal/mol above the C1- + CHpBr reactants asymptotic potential, which corresponds to approximately 13.4 kcal/mol of energy in the Cl----CHpBr complex. From the classical potential energies in Table 1 and using the harmonic frequencies in Table 2, the total energy available to the CICHp + B r products is then approximately 15.6 kcal/mol. The average thermal rotational energy at 300 K was included in the initial energy of CI----CHpBr by adding I/2RT to each principal axis of rotation. This corresponds to a total angular momentum of 91 h. A single normal mode of CI----CHpBr is excited for the initial conditions investigated here. For initial conditions in which zero-point energy is not added to CI----CHpBr, the energy for the excited normal mode is distributed randomly between potential and kinetic energy by choosing a random phase for the normal m0de.4~ If zero-point energy is first added to Cl----CHpBr, the procedure used to add vibrational energy to the complex depends on whether a CHpBr intramolecular mode or Cl----CH3Br intermolecular mode is excited. As described in previous work,22 the normal mode model is sufficiently accurate for adding zero-point energy to Cl----CHpBr and exciting a CHpBrintramolecular mode. As described above,in this model the energy for a normal mode is distributed randomly between potential and kinetic by choosing a random phase for the normal mode. However, the three intermolecular modes of the Cl----CH3Br complex (Le., one stretching and two bendings) are very anharmonic with low frequencies, and the above normal mode model, in which both potential and kinetic energies are added with random phase, is not sufficiently accurate for exciting an intermolecular mode with 13.4 kcal/mol of energy when zero-point energy is added. To overcome this problem, only kinetic energy was initially added when exciting an intermolecular normal mode by adding either positive or negative momentum to the mode. In this manner, the problem with anharmonicity was avoided and the normal mode model was maintained for studying selective initial excitation of the intermolecular modes. When exciting the intermolecular modes in this fashion, zeropoint energy was added to the CHpBr intramolecular modes with random phases as discussed above. Finally, when exciting two degenerate E modes like Cl----C bending, equal amounts of quanta are added to each of the degenerate modes. The trajectories were numerically integrated with combined fourthorder RungeKutta and sixth-order Adams-Moulton algorithms. An integration step size of 5 X 1 0 - l ' ~was used to ensure energy conservation to 5-6 significant numbers. The integration time limit for a single s (i.e. 25 ps). Either50or 100trajectories trajectorywasset to2.5 X were integrated for each specific mode excitation. The trajectories were analyzed for product ClCH3 + B r vibrational, rotational, and relative translational energies, lifetimes of the Cl----CHpBr and CICHp---Br complexes, and [Cl---CHp---Br]- central barrier crossings. Each trajectory was initialized in the Cl----CHsBr well and the time (Le., lifetime) the trajectory remained in this well was recorded. If the trajectory dissociated to C1- + CH3Br reactants, the lifetime was taken as the time of the last inner turning point in the CI- + CHpBr relative motion before dissociation. For isomerization to C1CH3---Br, the lifetime was taken as the time the trajectory passed the central barrier, where g, = ra - rb is given by g,' = 0.01 12 A. Once the central barrier (42) Hase, W. L.; Duchovic, R. J.; Hu, X.;Lim, K. F.; Lu, D.-h.; Swamy, K.N.;VandeLinde,S.R.; Wolf,R. J. VENUS, tobesubmittedtotheQuantum Chemistry Program Exchange. VENUS is an enhanced version of MERCURY: Hase, W. L. QCPE 1983, 3, 453. Until VENUS is submitted to QCPE, interested parties can receive a copy of VENUS by contacting the authors. (43) Sloane,C. S.;Hase, W. L. J. Chem. Phys. 1977,66,1523. Hase, W. L.; Ludlow, D. M.; Wolf, R. J.; Schlick, T. J. J . Phys. Chem. 1981, 85,958. (44) Zhu, L.; Hase, W. L. Chem. Phys. Lett. 1990, 175, 117. (45) Aubanel,E. E.; Wardlaw, D. M.; Zhu, L.; Hase, W. L. Int. Rev. Phys.
Chem. 1991, 10, 249. (46) Zhu, L.; Chen, W.; Hase, W. L.; Kaiser, E. W. J. Phys. Chem. 1993, 97, 311.
was crossed the trajectory was analyzed to determine whether it p r d e d directly to ClCHp + B r products, without an inner turning point in the ClCH, + B r relative motion. If not, the lifetime of the resulting CICHp--Br complex was recorded as described above for the CI----CHpBr complex.
IV. RRKM, PST,and OTS/PST Calculations To assist in interpreting the trajectory results, RRKM theory3438was used to calculate the unimolecular rate constants for the CI----CH,Br and ClCHp---Br complexes, and OTS/PST t h ~ r y ' ~was . ~used to calculate the rate constant for ClCHp---Br ClCHp + B r dissociation and the ClCHp + B r product energy partitioning. Thevibrator transition state model was used for the RRKM cal~ulations.4~ The rotational degree of freedom associated with the K quantum number is assumed to be active so that the RRKM rate constant is written as46948349
-
(3)
where Eo is the potential energy of the transition state and E, is the symmetric top rotational energy. The variational version of RRKM theorys' is used to calculate the transition state properties for Cl---CHpBr CI- CHpBr and CICH3---Br ClCHp + B r dissociation. The path of steepest descent in mass-weighted Cartesian coordinatess2 and reaction path H a m i l t ~ n i a nwere ~ ~ used ~ ~ ~ to find properties along the reaction paths for these dissociations. The variational transition state, for the Cl----CHpBr F? ClCHp---Br isomerizations, is located at the saddle point for the central barrier. For the energy and angular momentum considered in this work (see the previous section), the harmonic RRKM rate constants are the following: kdk = 0.3 ps-1 for Cl----CHpBr c1+ CHpBr; kbm = 0.006 ps-1 for Cl----CHpBr ClCH3-Br; kdi( = 2 ps-1 for CICHp---Br- ClCHp + B r ; and k- = 0.0005 ps-I for ClCHp--Br Cl----CHpBr. The average lifetime for a complex is the inverse of the sum of its kd" and kh,,, rate constants. The resulting RRKM lifetimes of the pre- and postreaction complexes are 3 and 0.5 ps, respectively. Preliminarycal~ulations~~ indicate thatanharmonicity may lower these rate constants by as much as a factor of five. A flexible transition state modelsm has also been used in variational RRKM calculations. For CI----CHpCI CI- + CHpCl dissociation at 300 K, a calculation of this type, which implicitly includesa consideration of the variation of the reaction coordinate away from the center-of-mass separation distance, gives the same rate constant as determined with the vibrator/reaction path variational RRKM calculation performed In addition, for C1- + CHpBr Cl----CH,Br association in the temperature range of 100-1000 K, the vibrator/reaction path variational model of eq 3 gives rate constants nearly the same4' as those determined with the statistical adiabatic channel model (SACM)t2 the orbiting transition state microcanonical variational transition state model?, and the trajectory capture Thus, the vibrator/reaction path
-
+
-
-
-
-
-
(47) The calculations were performed with a general RRKM computer program; Zhu,L.; Hase,W. L., submittedto theQuantumChemistry Program Exchange. (48) Quack, M.; Troe, J. Ber. Bunsenges. Phys. Chem. 1974, 78,240. (49) Miller, W. H. J. Am. Chem. SOC.1979, 101,6820. (50) Steinfield, J. I. Molecules andRadiation; MIT Press: London, 1985. (51) Hase, W. L. Acc. Chem. Res. 1983, 16, 258. (52) Fukui, K. J . Phys. Chem. 1970, 74, 4161. (53) Miller, W. H.; Handy, N. C.; Adams, J. E. J . Chem. Phys. 1980,72, 99.
(54) Kato, S.; Morokuma, K.J . Chem. Phys. 1980,73,3900. Morokuma, K.; Kato, S.In PotentialEnergy Surface and Dynamics Calculations;Truhlar, D. G., Ed.; Plenum Press: New York, 1981; p 243. (55) Peslherbe, G. H.; Wang, H.; Hase, W. L. to be submitted for publication. (56) Wardlaw, D. M.; Marcus, R. A. J . Phys. Chem. 1986,90, 5383. (57) Wardlaw, D. M.; Marcus, R. A. Ado. Chem. Phys. 1988,70, Part I, 231. (58) Aubanel, E. E.; Wardlaw, D. M.J . Phys. Chem. 1989,93, 3117. (59) Klippenstein, S.J. J. Chem. Phys. 1991, 94, 6469. (60) Klippenstein, S. J. J . Chem. Phys. 1992, 96, 367. (61) Klippenstein, S.J. private communication. (62) Troe, J. Chem. Phys. Lett. 1985, 122, 425. (63) Chesnavich, W. J.; Su,T.; Bowers, M. T. J. Chem. Phys. 1980, 72, 2641. (64) Su, T.; Chesnavich, W. J. J . Chem. Phys. 1982, 76, 5183.
J. Am. Chem. SOC.,Vol. 116, No. 21, 1994 9641
Trajectory Studies of S,2 Nucleophilic Substitution Table 3. CICHp
+ B r Average Product Energies and Angular Momentum from PST, OTS/PST, and Classical Trajectoriesa OTS/PST
PST propertyb (Et) (E,) (Ev)
ti)
atom + diatom
atom t sphere
atom + diatom
atom + sphere
trajectory
2.9 f 2.5 2.9 f 2.5 9.8 f 3.3 42.7 f 20.8
2.7 f 2.4 4.1 & 2.8 8.8 f 3.3 34.8 f 13.3
3.0 f 2.5 2.8 f 2.5 9.8 f 3.3 42.4 f 20.6
2.8 f 2.4 4.0 f 2.8 8.8 f 3.3 34.7 f 13.2
1.4 f 1.3 0.87 f 0.98 13.3 & 1.6 16.3 f 11.2
The total product energy and angular momentum are 15.6 kcal/mol and 91 h, respectively. Average energies are given in kcal/mol and angular momentum in h. The uncertainties are one standard deviation of the distribution. variational RRKM model used here is expected to accurately represent CI----CHpBr and CICH3-Br dissociation. Quantum harmonic phase space theory (PST)pgand orbiting transition state/phase space theory (OTS/PST)a wereused to calculate the product ClCH3 + B r vibrational energy, E,, rotational energy, E,, and relative translational energy, Et. PST and OTS/PST are statistical theories, whichmakedifferent assumptionsaboutthenatureand roleofthepotential energy surface for the unimolecular reaction. PST assumes the reaction potential energy surface is of no importance in calculating product properties, and as a result, the available energy is distributed statistically between product translation, rotation, and vibration, given the constraints of total energy and angular momentum. Chesnavich and Bowers@ modified PST by assuming (1) an orbiting transition state, located at the centrifugal barrier, with a statistical distribution of energy, and (2) an isotropic long-range potential, so that the centrifugal potential at this transition state is transformed to product translation. The OTS/PST calculationsreported here utilize the isotropicion-induced dipole potential
(4) where a is the polarizability of CHpCl and q is the charge of Br. The experimental value65 of 4.43 A3 was used for a. Equations have not been reported for treating an atom-symmetric top system either with PST or OTS/PST.66 However, Chesnavich and Bowersa have shown that it is an excellent approximation, in product energy partitioning calculations, to represent an atom-symmetric top pair by an atom-sphere pair and this approach is used here, as it has been in another recent OTS/PST calculation for the ClCH3 + B r products.sb The rotational constant for the sphere model is a geometric mean of the rotational constants for CH3CI. Tocompare with the atom-sphere model calculations, calculations were also performed for a linear triatom diatom + atom model. The equations used for the PST and OTS/PST calculations are listed in refs 39 and 40, respectively, and are not given here. As shown in Table 3, PST and OTS/PST give nearly identical results. The principal differences in the results for the atom + diatom and atom + sphere models are in the CICH3 rotational energy and angular momentum. This is expected, since the triatom diatom + atom model has only two rotational degrees of freedom. As shown in Table 3, (E,) for this model is approximately two-thirds of the value for the sphere atom + sphere model. Theaverageenergies in Table 3 arecalculated using quantum harmonic state counting for the available energy of 15.6 kcal/mol and angular momentum of 91h. The average energies are somewhat different if classical state counting is used for the total energy of 39.7 kcal/mol, which includes the CICH3 zero-point energy. For OTS/PST and the atom diatom model the average values become (E,) = 8.4 kcal/mol, (E,) = 3.5 kcal/mol, and ( E t ) = 3.7 kcal/mol, where the zero-point energy has been removed from ( E ” ) . Of particular interest is that with this classical state counting ( E t )is 3.7 kcal/mol, instead of 3.0 kcal/mol found by quantum state counting. It is also of interest to compare the PST and OTS/PST rate constants for CICHI---Br ClCH3 + Br- dissociation with the RRKM rate constant given above. For the sphere atom + sphere model, both the PST and OTS/PST rate constants are 0.5 ps-l, while the PST and OTS/ PST rate constants are 2 ps-1 for the triatom atom + diatom model. In comparison, the vibrator RRKM model which treats the system as a symmetric top gives a rate constant of 2 P S - ~ . ~ ’The difference between the sphere and triatom models is due to their different treatments of the
-
-
-
+
-
-
-
(65) Radzig, A.A.;Smirnov, b. M. Reference Data on Atoms, Molecules, and Ions; Springer-Verlag: Berlin, 1985. (66) We are in the process of developing an algorithm for treatingsymmetric
top systems in PST and OTS/PST calculations.
reactive system’s moment of inertia. For the triatom model, the moment of inertia of the complex CICH3---Br and the product CICHp is that for the non-symmetricaxes. For the sphere model, it is taken as the geometric mean of the three principal moments of inertia for both the complex CICH3---Br and the product CICHp. The moment of inertia along the symmetric axis is an order of magnitude smaller than the other two moments of inertia for ClCHp and two orders of magnitude smaller for the complex. With respect to the triatom model, using this geometric mean has the effect of lowering the rotational energy at a certain J and, thus, increasing both the sum and density of states at that J . From the comparisons of moments of inertia, it is obvious that this effect is bigger for the complex than for the product. Thus, the density of states is increased more than the sum of states, and the sphere model’s rateconstant becomes lower than that for the triatom model. In their OTS/PST calculations, Graul and Bowerssbused a value for a,in eq 4, different than the experimental value of 4.43 A’ used here. They used a scaled a so that the CICH3 + B r collision rate calculated from the Langevin formula for ion-induced dipoles was the same as that determined from ADO theory, using the experimental polarizability. This gave a scaled a of 13.3 A3. Using this value of a,instead of 4.43 A3, has an insignificant effect on both the product energy partitioning and rate constant. This is because increasing a moves the orbiting transition state toward the product asymptotic limit, so that OTS/PST approaches PST. As shown in Table 3, with a = 4.43 A3 the OTS/PST and PST results are already nearly identical.
V. Trajectory Results An ambiguity in classical trajectory simulations is the treatment of zero-point energy (ZPE)and its possible aphysical flow.68 An approach proposed by BowmRn et al.69 and Miller et al.70 for constraining ZPE flow in trajectory simulations may destroy classicalphase space structures71important for classical/quantum cor respondence^.^^ Thus, it has been proposed7’ that the use of this approach be restricted to systems with ergodic dynamics. A second approach, proposed by Alimi et al.,73 is applicable to van der Waals typesystems, which have highand low frequency modes and modes whose character does not change during the course of the reaction. The criteria for applicability of these two ClCH3 approaches are not satisfied for the C1- CH3Br B r reaction. The classicalmotion is not ergodic and the character of the modes does change appreciably during the course of the reaction. Thus, for this work, trajectories were calculated with and without adding ZPE to the normal modes of Cl----CH3Br. The calculations with ZPE added are called quasiclassical t r a j e ~ t o r i e s .For ~ ~ the calculations with ZPE,hv/2 is first added
+
-
+
(67) If the complex’s density of states is calculated for a symmetric top instead of a spherical top, the PST and OTS/PST rate constants for the sphere atom + sphere model are 3.2 times larger and similar to that for the vibrator RRKM model. (68) Hase, W. L. J . Phys. Chem. 1986, 90, 365. Lu. D.-h.; Hase, W. L.
-
J. Chem. Phys. 1988,89, 6723. (69) Bowman, J. M.; Gazdy, B.;Sun, Q.J. Chem. Phys. 1989,91,2859. (70) Miller, W. H.; Hase, W. L.; Darling, C. L. J. Chem. Phys. 1989,91, 2863. (71) Peslherbe, G. H.; Hase, W. L. J. Chem. Phys. 1994, 100, 1179. (72) See the articles in: Hase, W. L. Intramolecular and Nonlinear Dynamics; Advances in Classical Trajectory Methods, Vol. 1; JAI Press: Greenwich, CT, 1992. (73) Alimi, R.; Garcia-Vela, A.; Gerber, R. B. J . Chem. Phys. 1992, 96, 2034. (74) Truhlar,D. G.; Muckerman,J.T. InAtomMolecule Collision Theory; Bernstein, R. B., Ed.; Plenum Press: New York, 1979; p 505.
Wang et al.
9648 J. Am. Chem. SOC.,Vol. 116, No. 21, 1994 Table 4. Number of Different Events without ZPE Added to the TraiectorieP _________________
-
no. of events remain in complex Ac ~~~
remain in complex Bd A B crossings form products mode excited vibrational energyb form reactants E, CI- bend 48 (0)C 0 (0)C 0 0 (W 12.35 (60) 2 (ole 0 AI, C-C1 str 12.48 (48) 43 (0) 0 (0) 0 (0) 7 (0) 0 E, C-H str 12.52 (1.33) 0 (0) 0 (0) 0 (0) 50 (0) 0 AI, C-H str 12.60 (1.40) 0 (0) 0 (0) 50 (0) 0 (0) 0 E, CHI rock 12.45 (4) 0 (0) 0 (0) 50 (0) 0 (0) 0 E, CH3 def 12.45 (3) 0 (0) 0 (0) 50 (0) 0 (0) 0 0 (0) 50 (0) AI, CH3 def 12.58 (3) 0 (0) 0 (0) 14 48 (21) AI, C-Br str 12.86 (9) 0 (0) l(1) l(1) 0 A total of 50 trajectories were integrated for each initial condition. The maximum integration time for each trajectory is 25 ps. Energies are in units of kcal/mol. Numbers in parentheses are the number of quanta added to the mode. In most cases, integral quanta were added to maintain the total energy (vibration + rotation) approximately 2.7 kcal/mol above that of reactants’. For the C-H stretching E and A modes, the frequencies are sufficiently high that fractionalquanta were added to fulfill the energy requirement. C Ais the Cl----CHoBr complex. B is the ClCHa---Brcomplex. e The number in parentheses is the number of trajectories of this type that have multiple crossings of the central barrier. A trajectory with an A B crossing followed by a B A crossing has multiple crossings.
-
-
Table 5. Number of Different Events for Quasiclassical Trajectoriesu
mode excited E, C1- bend AI, C-Cl str E, C-H str AI, C-H str E, CHI rock E, CH3 def AI, CH3 def AI, C-Br str
vibrational energyb 12.35 (60) 12.48 (48) 12.52 (1.33) 12.60 (1.40) 12.45 (4) 12.45 (3) 12.58 (3) 12.86 (9)
form reactants
form products
88 (0)C 100 (0)
2 (0)C
no. of events remain in complex AC
0 (0)
15 (6)
17 (4) 11 (13) 26 (12) 23 (16) 25 (16) 28 (23)
4 (3) 2 (1) 3 (3) 1(1) 5 (4) 1(1)
remain in complex Bd
6 (1)C 0 (0)
4 (1)’
0 (0) 64 (31) 13 (29) 58 (26) 66 (33) 51 (25) 63 (44)
8 (3) 13 (4) 10 ( 5 ) 13 (6) 8 (8)
A
-.
B crossings 8
0 159 183 169 191 164 236
a A total of 100 trajectories were integrated for each initial condition. The maximum integration time for each trajectory is 25 ps. Footnotes b-e are the same as those in Table 4.
to each mode of C1----CH3Br, and then 3/2RTthermal rotational energy and the mode specific vibrational excitation is added. For the calculations without ZPE, no zero-point energy is added to Cl----CH3Br before adding the rotational energy and vibrational excitation. The total harmonic Z P E of Cl----CH3Br is 24.33 kcal/mol. A. Without ZPE. Listed in Table 4 are the numbers ofdifferent events observed when Z P E is not added in the trajectory initial conditions. Most of the trajectories remain in the Cl----CH3Br complex. For the C-H stretch, CH3 rock, and CH3 deformation excitations all the trajectories remain in this complex. Extensive reaction is only observed for the Cl----C stretch intermolecular mode and C-Br stretch intramolecular mode excitations. Exciting the Cl----C stretch intermolecular mode causes most of the complexes to dissociate to C1- CH3Br reactants. In contrast, exciting the C-Br stretch intramolecular mode leads to extensive Cl----CH3Br CICH3---Br isomerization. It is this excitation that leads to the single reactive event out of 400 total trajectories. The results in Table 4 show that without Z P E excitation the trajectory results are highly non-statistical and very mode specific. In some respects these results are not surprising, for without adding zero-point energy the density of states for the energy considered here is very small and much less than the quantum mechanical density of states.35 In the next section quasiclassical trajectory results are presented, for which Z P E is added to the Cl----CH3Br complex. This has the effect of increasing the total classical energy, which previous studies75 have shown tends to increase the coupling between modes and decrease the extent of mode specific dynamics. B. With ZPE. 1. Types of Trajectory Events. The different types of events observed in the trajectories with Z P E added to Cl----CH3Br are enumerated in Table 5 . When either the Cl----C stretch or bend intermolecular modes are excited, the predominate event is dissociation to the reactants C1- CH3Br. The trajectories, which go to products, are direct events and do not recross the central barrier. In contrast, when the intramo-
-
+
+
(75)
Lu, D.-h.; Hase, W.L.J . G e m .Phys. 1989, 91, 7490.
+
lecular modes of CH3Br are excited, dissociation to form C1CH3Br becomes unimportant. Instead, the majority of the trajectories remain in the postreaction ClCH3---Br potential energy well after 25 ps of intramolecular motion. This is a substantial non-RRKM result. The R R K M calculations presented in the previous section, for the energy and angular momentum considered here, give lifetimes of 3 and 0.5 ps for the Cl----CH3Br and CICH3---Br complexes, respectively. The above results show that the type of unimolecular and intramolecular dynamics exhibited by the Cl----CH3Br complex is differentiated by whether the intermolecular or intramolecular modes are initially excited. That this result does not depend on the specific inter- or intramolecular mode excited is rather strong evidence that one can identify two distinct types of Cl----CH3Br ion-dipole complexes, which are called intermolecular and
intramolecular complexes.22 With the CH3Br intramolecular modes of the CI----CHSBr complex initially excited, an important feature of the trajectories is the multiple crossings of the central barrier. As an example, nearly one-half of these trajectories, which remain in the postreaction ClCH,---Br complex B, have multiple central barrier crossings. For this to occur there must be a t least three crossings: B, B A, and then A B. If the trajectories were i.e. A integrated for longer than 25 ps, even more crossings would be expected to occur before the C1- CH3Br reactants or ClCH3 B r products are ultimately formed. The occurrence of these multiple crossings is a non-RRKM result, since the above R R K M calculations show that the ClCH3---Br complex should preferentially dissociate to ClCH3 B r products. The ratio of products/(A B crossings) is 0.12 for all the intramolecular mode excitation trajectories. If the trajectories were integrated for more than 25 ps, so that each trajectory attained either the B reactant or product asymptotic limit, the products/(A crossings) ratio may change. R R K M theory predicts 2/2.006 1 for this ratio. It is unlikely that including anharmonicity in the R R K M calculations for ClCH3---Br would decrease this ratio. Anharmonicity should be more important for the variational
- -
+
+
-
+
--
Trajectory Studies of S,2 Nucleophilic Substitution
J. Am. Chem. SOC., Vol. 116, No. 21, 1994 9649
Table 6. Lifetimes of the Ion-Dipole Complexes for Quasiclassical Trajectoriesa Cl----CH3Br after recrossin&) dissb isomc
initial Cl----CHsBr
mode excited E, C1- bend AI, C-CI str E, C-H str AI, C-H str E, CH3 rock E, CH3 def AI, CH3 def AI, C-Br str
energy 12.35(60) 12.48(48) 12.52 (1.33) 12.60 (1.40) 12.45 (4) 12.45 (3) 12.58 (3) 12.86 (9)
dissb
isomC
1.8 0.005
9.2
15.5 16.8
8.5 1.2 8.4 6.8 1.2 0.2
11.9
Lifetimes are in units of ps. Dissociation to C1Cl----CHpBr.
+ CH3Br.
+
-
+
+
1.5 0.2
9.5
1.3
0.6 0.8 0.5 0.4 0.6 0.8
6.8 8.3 8.0 8.1 8.1 9.6
2.2 2.8 2.3 2.2 2.2 2.4
The dissociation and isomerization lifetimes for the initially excited Cl----CH3Br complex strongly depend on the mode initially excited. When the Cl----C bend intermolecular modes are excited, the dissociation lifetime is similar to that of RRKM theory. However, exciting the Cl----C stretch intermolecular mode gives a dissociation lifetime three orders of magnitude shorter than the RRKM prediction. When the intramolecular modes are initially excited for the Cl----CH3Br complex, all the isomerization lifetimes, except that for C-Br stretch excitation, are - 5 times larger than the lifetime found when the C1----C bend intermolecular modeis excited. Exciting the C-Br stretch intramolecular mode gives a n isomerization lifetime of 0.2 ps, which is a n order of magnitude shorter than the RRKM lifetime. Cl----CH3Br complexes formed after B A recrossings have a much shorter lifetime for A B isomerization than do the initially prepared Cl----CH3Br complexes. Thus, there appears to be a correlation in the crossings such that, after a crossing occurs, there is a tendency for a n immediate recrossing. Except for the initial condition with C-Br stretch excitation, Cl----CH3Br complexes formed by B A isomerization have C1- CH3Br dissociation lifetimes similar to the RRKM value. The lifetimes for the ClCH3---Br postreaction complex are strikingly non-RRKM. RRKM theory predicts this complex to preferentially dissociate to ClCH3 B r with a lifetime less than 1 ps. In contrast, the trajectory lifetime for dissociation is approximately 10 ps. The trajectory lifetime for isomerization is 1-3 ps and significantly shorter than the trajectory time for dissociation, a result inconsistent with RRKM theory. A further analysis of the ClCH3---Br complex lifetimes was made by plotting the natural logarithm of the relative number of these complexes versus the complex lifetime. The time the central barrier is crossed for a n A B isomerization defines t = 0 for the complex. An analysis of this type was done for each of the intramolecular mode excitations in Table 5 and the resulting six plots of ln[N(t)/N(O)] versus t are very similar. The two mode excitations illustrated in Figure 2 are representative, where it is seen that the plots have two time constants; i.e. N ( t ) is biexponential. The average lifetime 7 for the short-time component varies from 0.6 to 1.2 ps for the six plots, while for the long-time component the lifetime varies from 5.3 to 8.3 ps. Also plotted in Figure 2 is the RRKM prediction for ln[N(t)/N(O)], which equals -(kisom kdis)t where kiWmand kdis are the ClCH3---Br C1----CH3Br and ClCH3---Br ClCH3 B r rate constants. At short time the trajectory and RRKM N ( t ) are in approximate agreement. However, this agreement is a t best fortuitous, since RRKM theory predicts dissociation to dominate, but the shorttime trajectory event is isomerization (see Table 5). The unimolecular dynamics of ClCH3---Br is strongly non-RRKM. 3. CICHs Br Product Energy Partitioning. The trajectories were analyzed for product relative translation, E,, rotation, E,, and vibration, E,, energies. Average values for these product energies were statistically the same for each of the intramolecular mode excitations. Thus, the energy partitioning results for the
- -
(76) Bhuiyan, L. B.;Hase, W. L. J . Chem. Phys. 1983, 78, 5052. (77) Hase, W. L.; Vande Linde, S.R. unpublished results.
10.8
Isomerization to ClCH,---Br. Dissociation to ClCH3 + B r . e Isomerization to
dissociation transition state than for the transition state a t the central barrier, since the former is looser with lower frequencies and contains more internal energye76This would make the RRKM ratio for products/(A B crossings) even closer to unity. In previous work,22 it has been suggested that the origin of ~ is weak coupling such central barrier recrossings in S Nreactions between intermolecular and intramolecular complexes. The intermolecular complex is strongly coupled to the dissociation channel, while the intramolecular complex can access the central barrier. That C1CH3---Br does not preferentially dissociate to ClCH3 B r products, but recrosses the central barrier, is strong evidence that there are both intramolecular and intermolecular ClCH3---Br complexes. Thus, central barrier recrossings arise from transitions between the C1----CH3Br and ClCH3---Br intramolecular complexes. A statistical model for these transitions would predict that the ratio ClCH3---Br/Cl----CH3Br equals the ratio of the densities of states for the ClCH3 and CH3Br intramolecular modes. This ratio is 4.4 for the total energy of the trajectories, and it is similar to the ratio of ClCH3---Br and Cl----CH3Br complexes which remain when the trajectories with initial Cl----CH3Br intramolecular mode excitations are halted after 25 ps of motion. The results in Table 4 give 6 for this latter ratio. With intramolecular mode excitations, approximately 40% of the trajectories which form products do so directly, with only one B barrier crossing. The remaining 60% of the reactive A trajectories have a t least two A + B barrier crossings. The fraction of the reactive trajectories, which are direct, is smallest for the A,, C-Br stretch intramolecular mode excitations. This is somewhat surprising, since exciting this mode in the Cl----CH3Br complex might be expected to give substantial direct substitution, as a result of the direct substitution that is observed for CiaCH3Clbcollisions with c-clb stretch However, there is very little reaction and central barrier recrossings are also important, when the c-clb stretch mode of the Cla----CH3c l b complex is excited.77 Clearly there are interesting questions concerning differences in the unimolecular dynamics of rotationally thermalized X----RY complexes and those formed by XRY collisions. 2. Lifetimes of the Ion-Dipole Complexes. Average trajectory lifetimes of the Cl----CH3Br (A) and ClCH3---Br (B) complexes with respect to dissociation and isomerization are listed in Table 6. A distinction is made between the lifetimes of the initially formed Cl----CH3Br complex and that formed by B A recrossings. In some cases the average lifetimes are rather qualitative as a result of the small number of events. To assist in analyzing the lifetimes in Table 6, it is useful to consider the above RRKM lifetimes, which are 3 ps for Cl----CH3Br and 0.5 ps for ClCHp---Br. It is straightforward to show that RRKM theory predicts the same average lifetimes for isomerizing and dissociating complexes.
-
3.4 5.3 4.2 2.1
ClCHo---Br dissd isome
- -
-
+
+
+
+
+
-
+
Wang et al.
9650 J . Am. Chem. SOC.,Vol. 116, No. 21, 1994 0.0
1.0
0.8
2
ea
0.6
0.4 0.2
0.0
1.0
-5.0
,
.
',
.
,
,
.
.
,
.
,
0.8 h
0.0
4.0
8.0
12.0
16.0
'
20.0
0.6
0.4
TIME (PSI
-
Figure 2. Relative number of ClCHp---Br complexes versus time. The complexes are formed by Cl----CH3Br ClCH,---Br central barrier crossing and decompose by either forming the ClCHp + B r products or recrossing the central barrier to form the Cl----CHpBr complex: (a) A,, CHp-deformation intramolecular mode excitation; and (b) E, CHp-rock intramolecular mode excitation. The solid lines are the fits to the shorttime and long-time decompositions. The dashed line is the RRKM prediction.
mode excitation trajectories were combined to give the distributions in Figure 3. The average values for these distributions are ( E , ) = 1.4 kcal/mol, ( E ; ) = 0.9 kcal/mol, and (Ev)= 13.3 kcal/mol, where ClCHp zero-point energy has been removed from the vibrational energy. The product energy is preferentially partitioned to ClCH3 vibration. The ClCHp average rotational energy is approximately two-thirds of the average relative translational energy. A comparison of the trajectory, PST, and OTS/PST average energies in Table 3 shows that PST and OTS/PST predict a higher relative translational energy and a lower vibrational energy than found from the trajectories. Product energy distributions determined by the trajectories and by OTS/PST for the atom sphere model are compared in Figure 3. It is worth emphasizing the significance of finding a trajectory relative translational energy distribution which is colder than that of quantum harmonic OTS/PST theory. If the trajectory dynamics were truly ergodic, the zero-point energy of the system would be free to redistribute and the total energy available for distribution in the products would be 39.7 kcal/mol. As discussed above in Section IV, classical OTS/PST theory predicts an average product relative translational energy of 3.7 kcal/mol. The energy partitioning observed from the trajectories shows that theclassical motion is not ergodic and instead is approximately conserving zero-point energy for the CH3Cl product. However, some "leak" of zero-point energy to rotation and translation is expected to occur.6s~7sAs a result, a quantum dynamical calculation may give lower product relative translational energies than found from the trajectory calculations. In their experimental study of Cl;---CH3Br dissociation, Graul and Bowerss also found that the ClCH, Br- product relative
+
+
0.2 0.0 0.0
2.0
4.0
6.0
8.0
10.0
12.0 14.0
16.0
Energy (kcaYmo1) Figure 3. Product energy distributions found from the intramolecular
mode excitation trajectories (histograms) and OTS/PST for the atom + sphere model (solid lines): (a) ClCHp + B r relative translationalenergy; (b) ClCHp rotational energy; and (c) CICHp vibrational energy.
translational energies are less than the predictions of OTS/PST. However, a direct comparison cannot be made with their work for the following reasons: (1) the potential energy surface used in this work has a reaction exothermicity including harmonic zero-point energies of 12.27 kcal/mol, while the best experimental estimate of the exothermicity is 8-9 kcal/mol;* (2) the Cl---CHjBr complexes studied here are more energized than those of Graul and Bowers; and (3) the nature of the Cl----CHpBr vibrational/rotational states studied by Graul and Bowers is uncertain. An important step in making a comparison is to scale the total product energy for our trajectories and OST/PST calculations to match that of Graul and Bowers experiments, which is estimateds to approximately equal the reaction exothermicity. For our calculations the total product energy is 15.6 kcal/mol and, if 9 kcal/mol is assumed for the reaction exothermicity,our trajectory ( E t )= 1.4 kcal/mol should bescaled by 9/15.6 to give 0.8 kcal/mol. This number is in excellent agreement with ( E t ) = 0.7 f 0.2 kcal/mol reported by Graul and Bowers.
VI. Summary In the work presented here classical trajectory calculations are used to study the unimolecular and intramolecular dynamics of Cl----CH3Br complexes excited with 13.4 kcal/mol of internal energy and 91h angular momentum. Mode specific excitation is investigated in the trajectories, which are evaluated for 25 ps of internal motion. For the energy and angular momentum of the trajectories, RRKM theory predicts Cl----CHpBr to prefer-
Trajectory Studies of S2 , Nucleophilic Substitution
+
entially dissociate to C1- CH3Br. The harmonic R R K M lifetime for Cl----CH3Br is 3 ps and this lifetime is only 0.5 ps for C1CH3---Br complexes formed by Cl----CH3Br ClCH3---Br isomerization. There are significant differences between trajectories calculated with and without ZPE added in the initial conditions. Without ZPE the reaction dynamics is highly mode specific and there is very little energy flow between modes. A similar type of dynamics was observed in trajectory studies of intramolecular vibrational energy redistribution in benzene without ZPE added.’S The principal qualitative differences between the trajectories with and without ZPE are as follows: (1) without Z P E there is negligible energy transfer between the CH3Br intramolecular modes of the Cl----CH3Br complex and central barrier crossing only occurs when the C-Br stretch mode is excited; and (2) very little ClCH3 B r product formation is observed in the trajectories without ZPE. Comparisons between classical and converged quantum calculations of C H stretch relaxation in benzene’s and linear hydrocarbon chains79 show that there is good agreement between the classical and quantum calculations if Z P E is included in the trajectory calculations. This result suggests that, for the properties investigated here for Cl----CH3Br decomposition, it is more appropriate to perform the trajectories with ZPE.sO The following are the important findings of this study for the trajectories with ZPE added: (1) The unimolecular dynamics of the Cl----CH3Br complex is highly mode specific. When the low-frequency Cl----C stretch and Cl----CH3Br bending modes of the complex are excited, the predominant event is dissociation to form C1-+ CH3Br. However, this is a negligible pathway when the higher frequency modes of ClCH3---Br the complex are excited. Instead, Cl----CH3Br isomerization occurs for >90% of these trajectories. Thedifferent dynamics exhibited by these two excitation patterns indicates there are two distinct types of C1----CH3Br complexes, which are called intermolecular and intramolecular complexes. Experiments to study the mode specific dynamics of these two complexes are planned.8l (2) The trajectories, which isomerize following CH3Br intramolecular mode excitation, do not immediately form ClCH3 B r as predicted by R R K M theory. Instead the majority undergo a t least one recrossing of the central barrier. After 25 ps of motion, approximately 25%of these trajectories have formed products, while more than one-half remain trapped in the ClCH3- - B r potential well. Presumably there is a substantial bottleneck for a transition from the ClCH3---Br intramolecular to intermolecular complex and these trajectories remain in the intramolecular complex. (3) Substantial recrossing of the central barrier is observed for trajectories with initial CH3Br intramolecular mode excitation.
-
+
-
+
(78) Wyatt, R. E.; Iung, C.; Leforestier,C. J . Chem. Phys. 1992,97,3477. (79) ToDaler. M.: Makri. N. J . Chem. Phvs. 1992. 97. 9001. (80) If irajectory calculations are to be usdd to determine fine dynamical details like the structure within a CH stretch absorption band, ref 77, or
lifetimes of long-lived resonance states, both of which arise from recurrences in the dynamics, it may be necessary to calculate the trajectorieswithout or withonly a small fraction of the ZPE. Including ZPE gives the broad dynamical features, but not fine structures in the dynamics. See discussions in: Lu, D.-h.;Hase, W. L J . Phys. Chem. 1988,92,3217. Gomez Llorente, J. M.; Hahn, 0.;Taylor, H. S.J . Chem. Phys. 1990, 92, 2762. Also ref 75. (81) Shin, S.K. private communication.
J. Am. Chem. Soc., Vol. 116, No. 21, 1994 9651
-
+
The ratio of ClCH3 B r product formation to C1----CH3Br ClCH3---Br crossings (Le. isomerizations) is approximately 0.1. This recrossing ratio may be different for other types of Cl---CH3Br excitations. Preliminary calculationssZ indicate that recrossings are less important than found in this work for C1---CH3Br complexes formed by C1- CH3Br association. (4) The energy distributions of the ClCH3 B r products formed from the trajectories are in substantial disagreement with the prediction of either PST or OTS/PST. In particular the trajectory relative translational and vibrational energies are lower and higher, respectively, than those of PST and OTS/PST. The form of the trajectory translational energy distribution is the same as measured by Graul and Bowerss for dissociating Cl---CH3Br complexes. If the reaction exothermicity of the trajectories is scaled to match that of the experiments, the theory and experimental energy distributions are nearly identical. However, the overall significance of this agreement is not clear, since it is unknown whether the experiments and trajectories are investigating the same types of dissociating states and the dependence of the product energy distribution of the nature of the dissociating state is unknown. In conclusion, it is useful to review what is known about the nature of the dynamics which gives rise to the nonstatistical behavior observed here for C1- CH3Br ClCH3 B r and CH3Clb C1,CHs Clb-. in previous ~ o r k l ~for- ~C1,-~ Trajectories for c1,- CH3Clb Cla----CH3Clb association show that the complex is formed by T R energy transfer and, initially, there is no energy transfer to the CH3Clb intramolecular modes, including the C X l b stretch. This latter mode must become excited ~ to occur and, thus, there is a dynamical for the S N reaction bottleneck for energy transfer from the low-frequency intermolecular modes of Cla----CH3Clb to the higher-frequency c-clb stretch. Such bottlenecks and weak couplings for energy flow have been found for numerous van der Waals systems,S3 whose potential energy surfaces are similar to those for the Cla----CH3clb, Cl----CH3Br, and ClCH3---Br ion-molecule complexes. By microscopic reversibility, if there is inefficient energy transfer from the intermolecular to intramolecular modes of the complex, energy transfer will also be inefficient for the reverse process. As a result, energy may remain trapped in the intramolecular modes, with concomitant central barrier recrossings and non-RRKM dynamics, as observed in the trajectories for Cla----CH3C1b ClaCH3---C1b-and Cl----CH3Br ClCH3---Br. A better model for dissociation of a S N 2 complex to form ion molecule products may be a nonstatistical vibrational predissociation mechanism,84 instead of the statistical energy transfer associated with R R K M theory. Additional work is currently in progress to provide more details for the above model. Of interest is the nature of the mode couplings and the phase space structure as the system descends the S N central ~ barrier and forms products (or reactants). A goal of this analysis is to determine which modes affect central barrier recrossings and contribute to couplings between the intermolecular and intramolecular ion-dipole complexes.
+
+
+ +
--
-
+
--.
+
+
-
+
Acknowledgment. This research was supported by the National Science Foundation. (82) Wang, H.; Hase, W. L. to be submitted for publication. (83) Miller, R. E. Science 1988, 240, 447. (84) Ewing, G. E. J. Chem. Phys. 1980, 72, 2096.
